Increasingly strict statutory regulations regarding admissible pollutant emissions in motor vehicles, in which internal combustion engines are disposed, make it necessary to keep the pollutant emissions during operation of the internal combustion engine as low as possible. This may be done firstly by reducing the pollutant emissions that arise during combustion of the air-fuel mixture in the respective cylinders. Secondly, in internal combustion engines use is made of exhaust gas after-treatment systems, by means of which the pollutant emissions produced during the process of combustion of the air-fuel mixture in the respective cylinder are converted into harmless substances. Especially in gasoline internal combustion engines, three-way catalytic converters are used as catalytic converters. In this connection it is necessary to ensure that the components of the exhaust gas after-treatment system also function in the desired manner over a long operating period and that faults are reliably detected.